One of the primary goals of Fourier Transform Mass Spectrometry (FTMS) is the identification of the ionic species, along with their relative abundances present, in a form of coherently oscillating ion packets contained by the trapping field within a mass spectrometer. The frequency of oscillation of a coherent packet of ions is a function of the mass to charge (m/z) ratio of the ionic species and is referred to herein as the “characteristic frequency” of an ionic species. The trapping field can be provided by the combination of an electrostatic field and a magnetostatic field, for example in a Fourier Transform Ion Cyclotron Resonance (FTICR) mass analyzer, or by an electrostatic field only, for example in an ORBITRAP mass analyzer. FTMS using RF fields is also known.
Typically, ions are detected by an image current S(t) (also termed a continuous transient image current and herein referred to as the “transient”) induced on detection electrodes of the mass analyzer as the oscillating ions pass nearby. Therefore, the transient comprises a superposition of one or more periodic signals. Each periodic signal corresponds to the oscillation of a respective coherent packet of ions within the mass analyzer with a respective characteristic frequency. The transient is only measured (or captured or recorded) over a finite time T, termed the “duration” of the transient.
The transient processing usually involves discrete Fourier transform (DFT), which decomposes the transient into a number of periodic functions (also termed Fourier basis functions). Each Fourier basis function is localized at a respective frequency (also termed a Fourier Transform bin). The frequencies corresponding to the Fourier basis functions form a set of frequencies (referred to as the Fourier grid). The Fourier basis functions are equally spaced in the frequency domain i.e. the separation between adjacent frequencies is a constant. In particular, the separation between adjacent frequencies in the set of frequencies (herein referred to as the “separation” of the set of frequencies) is determined by the inverse of the duration of the transient 1/T. The decomposition comprises calculating, based on the transient, individual complex amplitudes corresponding to each Fourier basis function. Thereby a set of complex amplitudes is formed. Therefore, the discrete Fourier transform (DFT) represents the transient in the frequency domain. In particular, the transient is represented as a set of complex amplitudes. Each complex amplitude of the set of complex amplitudes corresponds to a respective frequency of the set of frequencies i.e. the frequency at which the corresponding Fourier basis function is localized.
The periodic signals present in the transient (as described previously) are related to the complex amplitudes. In particular, the periodic signal will contribute to the complex amplitudes corresponding to a plurality of frequencies in the set of frequencies. The plurality of frequencies will be substantially centred on the characteristic frequency of a particular ionic species for given experimental conditions. Therefore a plot of the set of complex amplitudes against the set of frequencies (referred to as a mass spectrum) will show one or more peaks, each peak substantially centred on a respective characteristic frequency present in the transient i.e. the centroid of each peak will be substantially equal to the characteristic frequency.
As described above, the frequencies of the periodic signals present in the transient are a function of the m/z ratios of the ionic species. Therefore, the centroid of each peak can be converted (or transformed or interpreted) into a respective m/z ratio thereby identifying a respective ionic species. Furthermore the height of each peak can be converted (or transformed or interpreted) into the respective relative abundance of the respective ionic species.
Due to the spacing of frequencies in the Fourier grid, determining the centroids of the peaks, and/or the heights of the peaks can be subject to errors. These errors lead to errors in the estimation of correct m/z ratios (and therefore ionic species being identified incorrectly) along with errors in the estimation of relative abundances. These errors can be particularly significant when the difference between a characteristic frequency present in the transient and the closest frequency in the set of frequencies is large.
A number of approaches aimed at apparent “smoothing” of spectra (hence potentially reducing the estimation errors) currently employed include interpolating the complex amplitudes onto a further set of frequencies with reduced separation between frequencies i.e. interpolating the mass spectrum. The most common interpolation method is zero-padding (see Marshall A. G.; Verdun, F. R., “Fourier Transforms in NMR, Optical, and Mass Spectrometry”, Elsevier, 1990, the entire contents of which are incorporated herein by reference). Zero padding, if done explicitly, comprises of appending the transient by zero-signal of a predetermined duration resulting in an artificial increase of the transient duration and, correspondingly a decrease in the separation of the set of frequencies. Therefore, if the transient duration is increased by a factor P through the appending of a zero-signal, the separation of the set of frequencies is correspondingly reduced by a factor P. Due to the mechanics of implementation of Fast Fourier Transform (FFT), the most common algorithm for computing DFT, it is common for P to be a power of two and the interpolated mass spectrum is called log2 P-times zero-padded.
Whilst this can reduce the errors described above in relation to isolated peaks corresponding to respective characteristic frequencies, there is still a problem when a transient comprises two or more close characteristic frequencies. This causes the spectrum to comprise two or more overlapping peaks. If the separation (or difference) between two characteristic frequencies of the transient is less than a threshold value, then the two peaks will not be resolved. This error leads to errors in the converted m/z ratios (and therefore ionic species being identified incorrectly) along with errors in the converted relative abundances. Although it depends on the local spectral density, the practical threshold value for reliable resolution is twice the separation of the Fourier grid corresponding to the original transient i.e. the transient without zero padding.
FIG. 1A of the accompanying drawings shows an example of such a problem. The figure shows a first signal 150 of a transient, a second signal 160 of the transient and a spectrum 170 of the transient. The first signal 150 has a characteristic frequency f1. The second signal 160 has a characteristic frequency f2. The difference between f1 and f2 is equal to the separation of the Fourier grid. The spectrum 170 has two central peaks. The leftmost peak of the spectrum 170 corresponds to the second signal 160. The rightmost peak of the spectrum 170 corresponds to the first signal 150. There is also an error 174 between the centroid of the peak and the associated characteristic frequency. There is an error 172 between the height (or intensity) of the peak and the height (or intensity) of the corresponding signal 150, 160. The errors become more pronounced as the spectral density (i.e. number of harmonic components for a given region of a spectrum) increases, and their separation diminishes.
FIG. 1B of the accompanying drawings illustrates the problem. The figure shows a first signal 150 of a transient, a second signal 160 of the transient and a spectrum 170 that will be reproduced from the transient. In this case, the difference between f1 and f2 is equal to half the separation of the Fourier grid. The spectrum 170 has a single peak i.e. the characteristic frequencies corresponding to the two signals 150, 160 are not resolved. The centroid of the single peak of the spectrum is in error compared to either of the two characteristic frequencies. Additionally, the height of the single peak is neither equivalent to the sum of the heights of the two signals 150, 160 nor either one of the heights of the two signals 150, 160. Due to these errors, neither of the ionic species corresponding to the signals 150, 160 will be correctly identified. Also the relative abundance reported from the peak will be incorrect. This may lead to errors in abundance ratios calculated using other peaks in the signal 170 which may, themselves be accurate.
Interpolation of the spectrum, for example by zero-padding as described above, neither reduces these errors nor improves the resolution. In fact, the zero-padded and optionally apodized FT amplitudes are linear combinations of the FT amplitudes and carry no extra useful information. This can be seen by the fact that the complex amplitudes sn of the Fourier transform of a signal S(t) without zero-padding obey the following relation:
      s    n    =            1      T        ⁢                  ∫        0        T            ⁢                        S          ⁡                      (            t            )                          ⁢                  exp          ⁡                      (                                          -                2                            ⁢              π              ⁢                                                          ⁢              i              ⁢                              n                T                            ⁢              t                        )                          ⁢        d        ⁢                                  ⁢        t            whilst the complex amplitudes bm of the Fourier transform of the signal S(t) with P times zero-padding obey the following relation:
      b    m    =            1      T        ⁢                  ∫        0        T            ⁢                        S          ⁡                      (            t            )                          ⁢                  exp          ⁡                      (                                          -                2                            ⁢              π              ⁢                                                          ⁢              i              ⁢                              m                PT                            ⁢              t                        )                          ⁢        d        ⁢                                  ⁢        t            
Therefore, the complex amplitudes bm (i.e. with the zero-padding) are required to be linear combinations of the complex amplitudes sn (i.e. without the zero-padding). In particular, the complex amplitudes bm must obey:
      b    m    =            1      T        ⁢                  ∑        n            ⁢                        ∫          0          T                ⁢                              exp            ⁡                          (                              2                ⁢                π                ⁢                                                                  ⁢                i                ⁢                                  n                  T                                ⁢                t                            )                                ⁢                      exp            ⁡                          (                                                -                  2                                ⁢                π                ⁢                                                                  ⁢                i                ⁢                                  m                  PT                                ⁢                t                            )                                ⁢          d          ⁢                                          ⁢          t          ⁢                                          ⁢                      s            n                              which reduces to:
      b    m    =      P    ⁢                            exp          ⁡                      (                                          -                2                            ⁢              π              ⁢                                                          ⁢              i              ⁢                              m                P                                      )                          -        1                    2        ⁢        π        ⁢                                  ⁢        i              ⁢                  ∑        n            ⁢                        s          n                          nP          -          m                    